Underwater optical wireless communication (UOWC) has recently gained attention as an emerging solution to the growing demand for high-speed underwater wireless communications. However, the performance of UOWC networks could be highly affected by the multi-node interference, especially in three-dimensional (3-D) network configuration. Characterizing the light propagation through an underwater channel that includes the interference effect is crucial to performance analysis and improvement of the networks. Taking into account this effect and the 3-D UOWC network configuration, we derive in this paper an approximation of the signal-to-interference-plus-noise ratio (SINR), which is a fundamental performance metric for design of network protocols. In the numerical results, Monte Carlo simulations are performed to validate the accuracy of theoretical derivation. Also, we show the dependence of SINR on various network factors such as water types, number of nodes, communication distance and transmission power.